Solve for x (complex solution)
x=\frac{13+\sqrt{2351}i}{20}\approx 0.65+2.424355584i
x=\frac{-\sqrt{2351}i+13}{20}\approx 0.65-2.424355584i
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10x^{2}-13x+63=0
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 10\times 63}}{2\times 10}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -13 for b, and 63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-13\right)±\sqrt{169-4\times 10\times 63}}{2\times 10}
Square -13.
x=\frac{-\left(-13\right)±\sqrt{169-40\times 63}}{2\times 10}
Multiply -4 times 10.
x=\frac{-\left(-13\right)±\sqrt{169-2520}}{2\times 10}
Multiply -40 times 63.
x=\frac{-\left(-13\right)±\sqrt{-2351}}{2\times 10}
Add 169 to -2520.
x=\frac{-\left(-13\right)±\sqrt{2351}i}{2\times 10}
Take the square root of -2351.
x=\frac{13±\sqrt{2351}i}{2\times 10}
The opposite of -13 is 13.
x=\frac{13±\sqrt{2351}i}{20}
Multiply 2 times 10.
x=\frac{13+\sqrt{2351}i}{20}
Now solve the equation x=\frac{13±\sqrt{2351}i}{20} when ± is plus. Add 13 to i\sqrt{2351}.
x=\frac{-\sqrt{2351}i+13}{20}
Now solve the equation x=\frac{13±\sqrt{2351}i}{20} when ± is minus. Subtract i\sqrt{2351} from 13.
x=\frac{13+\sqrt{2351}i}{20} x=\frac{-\sqrt{2351}i+13}{20}
The equation is now solved.
10x^{2}-13x+63=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
10x^{2}-13x+63-63=-63
Subtract 63 from both sides of the equation.
10x^{2}-13x=-63
Subtracting 63 from itself leaves 0.
\frac{10x^{2}-13x}{10}=-\frac{63}{10}
Divide both sides by 10.
x^{2}-\frac{13}{10}x=-\frac{63}{10}
Dividing by 10 undoes the multiplication by 10.
x^{2}-\frac{13}{10}x+\left(-\frac{13}{20}\right)^{2}=-\frac{63}{10}+\left(-\frac{13}{20}\right)^{2}
Divide -\frac{13}{10}, the coefficient of the x term, by 2 to get -\frac{13}{20}. Then add the square of -\frac{13}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{13}{10}x+\frac{169}{400}=-\frac{63}{10}+\frac{169}{400}
Square -\frac{13}{20} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{13}{10}x+\frac{169}{400}=-\frac{2351}{400}
Add -\frac{63}{10} to \frac{169}{400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{13}{20}\right)^{2}=-\frac{2351}{400}
Factor x^{2}-\frac{13}{10}x+\frac{169}{400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x-\frac{13}{20}\right)^{2}}=\sqrt{-\frac{2351}{400}}
Take the square root of both sides of the equation.
x-\frac{13}{20}=\frac{\sqrt{2351}i}{20} x-\frac{13}{20}=-\frac{\sqrt{2351}i}{20}
Simplify.
x=\frac{13+\sqrt{2351}i}{20} x=\frac{-\sqrt{2351}i+13}{20}
Add \frac{13}{20} to both sides of the equation.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
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Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}